One classical and difficult problem in algebraic number theory is the determination of the rank of the \(p\) -class group of a number field \(K\) . When \(p=2\)  and \(K\)  is a quadratic extension of a number field \(k\)  having an odd class number, the ambiguous class number formula can be used to determine this rank, involving units of \(k\)  which are norms in \(K/k\)  and ramified primes in \(K/k\)  (cf. [1]}). This fact is practically one of the most important means for structuring the 2-class group of a given number field of small degree (cf. [2]}, [3]}). Our contribution in this article is
to study the 2-rank of an infinite family of number fields, with large degree over \(\mathbb {Q}\) . Comparing with other papers tackling this problem, the main novelty of this article is the combination of ramification theory, ambiguous class number formula and the theory of
cyclotomic \(\mathbb {Z}_2\) -extensions of some number fields.
